mlsanityData to 7 Oct 2026

Partial Differential Equations Meet Deep Neural Networks: A Survey

arXiv 2211.05567 ↗First posted 27 Oct 2022cs.LG cs.AIcited by 25

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Abstract (arXiv)

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics such as computational fluid dynamics, multiphysics simulation, molecular dynamics, or even dynamical systems. It is a vibrant multi-disciplinary field of increasing importance and with extraordinary potential. At the same time, solving PDEs efficiently has been a long-standing challenge. Generally, except for a few differential equations for which analytical solutions are directly available, many more equations must rely on numerical approaches such as the finite difference method, finite element method, finite volume method, and boundary element method to be solved approximately. These numerical methods usually divide a continuous problem domain into discrete points and then concentrate on solving the system at each of those points. Though the effectiveness of these traditional numerical methods, the vast number of iterative operations accompanying each step forward significantly reduces the efficiency. Recently, another equally important paradigm, data-based computation represented by deep learning, has emerged as an effective means of solving PDEs. Surprisingly, a comprehensive review for this interesting subfield is still lacking. This survey aims to categorize and review the current progress on Deep Neural Networks (DNNs) for PDEs. We discuss the literature published in this subfield over the past decades and present them in a common taxonomy, followed by an overview and classification of applications of these related methods in scientific research and engineering scenarios. The origin, developing history, character, sort, as well as the future trends in each potential direction of this subfield are also introduced.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). Builds on 82 of its 257 references that are in the map. First posted on arXiv 27 Oct 2022.

Builds on

82 of 257 references.

References · 82

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38 more
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 5; 2024: 5; 2025: 6; 2026: 9.

25 papers.

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  1. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  2. 2023From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyondcited by 26
  3. 2025Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modelingcited by 8
  4. 2023Efficient Neural PDE-Solvers using Quantization Aware Trainingcited by 6
  5. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  6. 2025Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operatorscited by 6
  7. 2024Physics-Informed Neural Networks with Trust-Region Sequential Quadratic Programmingcited by 4
  8. 2024GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equationscited by 2
  9. 2025Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solverscited by 2
  10. 2026Autoregression-Free Neural Operators for Time-Dependent PDEscited by 2
  11. 2023GNRK: Graph Neural Runge-Kutta method for solving partial differential equationscited by 1
  12. 2024A Numerical Study of Chaotic Dynamics of K-S Equation with FNOscited by 1
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  1. 2026Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equationscited by 1
  2. 2023Application of Tensor Neural Networks to Pricing Bermudan Swaptionscited by 0
  3. 2023Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networkscited by 0
  4. 2025Learning thermodynamic master equations for open quantum systemscited by 0
  5. 2025Operator Learning for Consolidation: An Architectural Comparison for DeepONet Variantscited by 0
  6. 2025AI paradigm for solving differential equations: first-principles data generation and scale-dilation operator AI solvercited by 0
  7. 2026Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problemscited by 0
  8. 2026Efficient Approximation to Analytic and Lp functions by Height-Augmented ReLU Networkscited by 0
  9. 2026Coordinate Encoding on Linear Grids for Physics-Informed Neural Networkscited by 0
  10. 2026Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogatescited by 0
  11. 2026Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equationscited by 0
  12. 2026One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEscited by 0
  13. 2026Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transportcited by 0